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Should Scoring Rules be "Effective"?


  • Robert F. Nau

    (A. B. Freeman School of Business, Tulane University, New Orleans, Louisiana 70118)


A scoring rule is a reward function for eliciting or evaluating forecasts expressed as discrete or continuous probability distributions. A rule is strictly proper if it encourages the forecaster to state his true subjective probabilities, and effective if it is associated with a metric on the set of probability distributions. Recently, the property of effectiveness (which is stronger than strict properness) has been proposed as a desideratum for scoring rules for continuous forecasts, for reasons of "monotonicity" in keeping the forecaster close to his true probabilities, since in practice the forecast must be chosen from a low-dimensional set of "admissible" distributions. It is shown in this paper that what effectiveness implies, beyond strict properness, is not a monotonicity property but a transitivity property, which is difficult to justify behaviorally. The logarithmic scoring rule is shown to violate the transitivity property, and hence is not effective. The L 1 and L \infty metrics are shown to allow no effective scoring rules. Some potential difficulties in interpreting admissible forecasts are also discussed.

Suggested Citation

  • Robert F. Nau, 1985. "Should Scoring Rules be "Effective"?," Management Science, INFORMS, vol. 31(5), pages 527-535, May.
  • Handle: RePEc:inm:ormnsc:v:31:y:1985:i:5:p:527-535

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    Cited by:

    1. Borgonovo, Emanuele & Marinacci, Massimo, 2015. "Decision analysis under ambiguity," European Journal of Operational Research, Elsevier, vol. 244(3), pages 823-836.
    2. Natalia Nolde & Johanna F. Ziegel, 2016. "Elicitability and backtesting: Perspectives for banking regulation," Papers 1608.05498,, revised Feb 2017.
    3. Lambert, Nicolas S. & Langford, John & Wortman Vaughan, Jennifer & Chen, Yiling & Reeves, Daniel M. & Shoham, Yoav & Pennock, David M., 2015. "An axiomatic characterization of wagering mechanisms," Journal of Economic Theory, Elsevier, vol. 156(C), pages 389-416.
    4. Freddy Delbaen & Fabio Bellini & Valeria Bignozzi & Johanna F. Ziegel, 2016. "Risk measures with the CxLS property," Finance and Stochastics, Springer, vol. 20(2), pages 433-453, April.
    5. Werner Ehm & Tilmann Gneiting & Alexander Jordan & Fabian Kr├╝ger, 2016. "Of quantiles and expectiles: consistent scoring functions, Choquet representations and forecast rankings," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(3), pages 505-562, June.
    6. Nolan Miller & Paul Resnick & Richard Zeckhauser, 2005. "Eliciting Informative Feedback: The Peer-Prediction Method," Management Science, INFORMS, vol. 51(9), pages 1359-1373, September.


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